A particle of mass $m_1$ elastically collides with another stationary particle of mass $m_2.$ Then if

$N$ beads identical beads are resting on a smooth horizontal wire which is circular at the end with radius $r=0.5 \mathrm{m}$ as shown in the figure. Find the minimum velocity which should be imparted to the first bead such that $n^{\mathrm{th}}$ bead will fall in the tank after completing full circle in vertical plane as shown in the figure. Take all the collisions between the beads elastic $(e=1).$

An elevator platform is going up at a speed $20{ \mathrm{ms}}^{-1}$ and during its upward motion a small ball of $50 \mathrm{g}$ mass falling in downward direction strikes the platform elastically at a speed $5{ \mathrm{ms}}^{-1}.$ Find the speed (in ${\mathrm{ms}}^{-1}$) with which the ball rebounds.

A ball of mass $1 \mathrm{kg}$ moving with a velocity of $5 \mathrm{m}/\mathrm{s}$ collides elastically with rough ground at an angle $\theta$ with the vertical as shown in figure. What can be the minimum coefficient of friction if ball rebounds vertically after collision? (given $\tan \theta =2$)

Three balls $A, B$ and $C\left(m_A=m_C=4m_B\right)$ are placed on a smooth horizontal surface. Ball $B$ collides with ball $C$ with an initial velocity $v$ as shown in figure. Find the total number of collisions between the balls (all collisions are elastic).

A particle with a mass of $1 \mathrm{kg}$ is having a velocity of $10 \mathrm{m}/\mathrm{s}$ in +ve x -direction at $t=0.$ Forces ${\overrightarrow{F}}_1$ and ${\overrightarrow{F}}_2$ act on the particle whose magnitudes are changing with time according to the variation shown in figure. The magnitude of the velocity of the particle at $t=3\mathrm{s}$ (neglect gravity effect) is found to be $n\sqrt{5} \mathrm{m}/\mathrm{s}.$. Find the value of $n.$

Two identical smooth balls are projected from points O and A on the horizontal ground with same speed of projection. The angle of projection in each case is $30°$ (see figure). The distance between O and A is $100 \mathrm{m}.$ The balls collide in mid-air and return to their respective points of projection. If the coefficient of restitution is $0.7,$ find the speed of projection of either ball (in $\mathrm{m}/\mathrm{s}$) correct to nearest integer. (Take $g=10 \mathrm{m}{ \mathrm{s}}^{-2}$ and $\sqrt{3}=1.7$)